A packing carton slides down an inclined plane of angle 30.0^∘ and of incline length 2.0 m. (a)

A packing carton slides down an inclined plane of angle...

A packing carton slides down an inclined plane of angle $30.0^{\circ}$ and of incline length $2.0 \mathrm{m}$. (a) If the initial speed of the carton is $4.0 \mathrm{m} / \mathrm{s}$ directed down the incline, what is the speed at the bottom? Ignore friction. (b) How long does it take the carton to slide down the incline?

A packing carton slides down an inclined plane of angle $30.0^{\circ}$ and of incline length $2.0 \mathrm{m}$. (a) If the initial speed of the carton is $4.0 \mathrm{m} / \mathrm{s}$ directed down the incline, what
is the speed at the bottom? Ignore friction. (b) How long does it take the carton to slide down the incline?

Key Concepts

Inclined Plane Dynamics

This concept involves analyzing motion along a sloping surface where forces such as gravity must be resolved into components parallel and perpendicular to the surface. Understanding inclined plane dynamics is key to solving problems where objects accelerate down slopes, allowing us to determine net acceleration and other kinematic quantities.

Component of Gravitational Force

When dealing with motion on an incline, the gravitational force acting on an object is decomposed into two components: one perpendicular to the incline and one parallel. The component parallel to the incline (mg sin(?)) causes the object to accelerate down the slope, which is critical for predicting the motion of objects in such scenarios.

Energy Conservation Principle

The conservation of mechanical energy principle states that the total energy (kinetic plus potential) remains constant in the absence of non-conservative forces like friction. This concept is useful in inclined plane problems to relate the initial kinetic and potential energies of an object to its final kinetic energy, thereby finding quantities such as final speed.

Kinematic Equations for Constant Acceleration

The kinematic equations describe the relationships among displacement, velocity, acceleration, and time for objects moving at a constant acceleration. These equations are crucial in inclined plane problems, as they allow for the calculation of final velocity or time of travel when the object moves under a constant net acceleration derived from gravitational forces.

Transcript

00:01 Pack of carton on the incline with the slope alpha and the length of the incline is l, which is 2 .0 meters.

00:13 So we have to find time when the pack of the carcin will reach the bottom.

00:20 So first let's show the height of the initial height of the carton above the ground, which is h0.

00:28 So obviously it equals to l multiply by sign alpha.

00:34 So now let's answer question a.

00:37 And first we have to write down the conservation of the energy.

00:41 Total mechanical energy is constant because there is no friction.

00:48 And it equals to the sum of kinetic and potential energies.

00:52 So it means that the initial kinetic energy plus initial potential energy equals to the final kinetic energy plus final potential energy.

01:00 Final potential energy is zero because the box reaches a ground.

01:09 Let's introduce some white axis which has a zero at bottom so and therefore final kinetic energy equals to the sum of the initial potential initial kinetic energy and initial final energy or mass of the particle multiplied by final kinetic energy squared divided by two equals to mass multiplied by initial potential initial velocity multiplied square divided by two plus mass multiplied by g multiplied by the initial height of the particle.

01:53 So it means that we can now calculate final velocity, which is square root of initial square velocity plus 2g multiplied by l multiplied by sine alpha.

02:11 So let's calculate this number...

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